Banach Journal of Mathematical Analysis

G-convergence and homogenization of monotone damped hyperbolic equations

Gabriel Nguetseng, Hubert Nnang, and Nils Svanstedt

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Multiscale stochastic homogenization is studied for quasilinear hyperbolic problems. We consider the asymptotic behaviour of a sequence of realizations of the form ${\frac{\partial^2 u^\omega_{\varepsilon}}{\partial t^2}} - \mathrm{div}\left(a\left(T_1(\frac{x}{\varepsilon_1})\omega_1, T_2(\frac{x}{\varepsilon_2})\omega_2 ,t, D u^\omega_{\varepsilon}\right)\right)-\Delta(\frac{\partial u^\omega_{\varepsilon}}{\partial t}) +G\left(T_3(\frac{x}{\varepsilon_3})\omega_3 ,t,\frac{\partial u^\omega_{\varepsilon}}{\partial t}\right)=f$. It is shown, under certain structure assumptions on the random maps $a\left(\omega_1,\omega_2,t,\xi\right)$ and $G\left(\omega_3,t,\eta\right)$, that the sequence $\{u^\omega_\epsilon\}$ of solutions converges weakly in $L^p(0,T;W^{1,p}_0(\Omega))$ to the solution $u$ of the homogenized problem ${\frac{\partial^2 u}{\partial t^2}} - \mathrm{div}\left( b \left( t,D u \right)\right)-\Delta(\frac{\partial u}{\partial t})+{\overline G}(t,\frac{\partial u}{\partial t}) = f$.

Article information

Banach J. Math. Anal., Volume 4, Number 1 (2010), 100-115.

First available in Project Euclid: 27 April 2010

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Zentralblatt MATH identifier

Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]
Secondary: 35B40: Asymptotic behavior of solutions

G-convergence multiscale stochastic homogenization elliptic parabolic hyperbolic


Nguetseng, Gabriel; Nnang, Hubert; Svanstedt, Nils. G-convergence and homogenization of monotone damped hyperbolic equations. Banach J. Math. Anal. 4 (2010), no. 1, 100--115. doi:10.15352/bjma/1272374674.

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  • V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff Publ., 1976.
  • A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, 1978.
  • A.C. Biazutti, On a nonlinear evolution equation and its applications, Nonlinear Anal. 24 (1995), no. 8, 1221–1234.
  • V. Chiado-Piat, G. Dal Maso and A. Defranceschi, $G$-convergence of monotone operators, Ann. Inst. H. Poincaré Anal. Non Linéare 7 (1990), no. 3, 123–160.
  • V. Chiado-Piat and A. Defranceschi, Homogenization of monotone operators, Nonlinear Anal. 14 (1990), no. 9, 717–732.
  • N. Dunford and J.T. Schwartz, Linear Operators Part 1 General Theory, Wiley, 1957.
  • Y. Efendiev and A. Pankov, Numerical homogenization of nonlinear random parabolic operators, Multiscale Model. Simul. 2 (2004), no. 2, 237–268.
  • V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, 1994.
  • G. Nguetseng, H. Nnang and N. Svanstedt, Asymptotic analysis for a weakly damped wave equation with application to a problem arising in elasticity, J. Funct. Spaces Appl., to appear.
  • G. Nguetseng, H. Nnang and N. Svanstedt, Spatial-reiterated homogenization of quasilinear hyperbolic equations in a general deterministic setting, Submitted.
  • H. Nnang, Un theoreme de existence et unicité, University of Yaounde, 2000, Preprint.
  • A. Pankov, $G$-convergence and Homogenization of Nonlinear Partial Differential Operators, Kluwer Publ., 1997.
  • S. Spagnolo, Convergence of Parabolic Equations, Boll. Un. Mat. Ital. B (5) 14 (1977), no. 2, 547–568.
  • N. Svanstedt, $G$-convergence of parabolic operators, Nonlinear Anal. 36 (1999), no. 7, 807–842.
  • N. Svanstedt, Multiscale stochastic homogenization of monotone operators, Netw. Heterog. Media 2 (2007), no. 1, 181–192.
  • N. Svanstedt, Convergence of quasi-linear hyperbolic operators,. Hyperbolic Differ. Equ. 4 (2007), no. 4, 655–677.